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Physical review. B, Condensed matter

Isotropic spin-1 chain with twisted boundary condition

Gábor Fáth, Jenő Sólyom

Published Jan 1, 1993• 73 citations
Publisher

Abstract

The possible ground states and the phase transitions of a spin-1 chain with bilinear-biquadratic exchange are studied numerically. In order to see better the real asymptotic behavior of the infinite system and to determine the necessary degeneracies, a twisted boundary condition is used that reduces the finite-size effects. This calculation confirms that a second-order Ising-type phase transition occurs at the Takhtajan-Babujian point from the dimerized phase to the singlet Haldane phase. The transition at the Lai-Sutherland point, to a massless phase with three soft modes, is found to be of Kosterlitz-Thouless type.

DOI
10.1103/physrevb.47.872

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The possible ground states and the phase transitions of a spin-1 chain with bilinear-biquadratic exchange are studied numerically. In order to see better the real asymptotic behavior of the infinite system and to determine the necessary degeneracies, a twisted boundary condition is used that reduces the finite-size effects. This calculation confirms that a second-order Ising-type phase transition occurs at the Takhtajan-Babujian point from the dimerized phase to the singlet Haldane phase. The transition at the Lai-Sutherland point, to a massless phase with three soft modes, is found to be of Kosterlitz-Thouless type.
Measurements done on LiVGe2O6 by Millet et al. [Phys. Rev. Lett. 83, 4176 (1999)] on phases of the Singlet ground state of the Spin-1 Chain can be described by the Bilinear-biquadratic spin-1 chain. I will discuss these results, among the results of others and talk about theoretical calculations done by A. Lauchli [Phys. Rev. B 74, 144426 (2006)] that describe the phase diagram of the bilinear-biquadratic model. At a particular point in the phase diagram the model becomes SU(3) globally symmetric and is known as the Lai-Sutherland model, it is Bethe ansatz solvable as shown in [A Schmitt et al., J. Phys. A: Math. Gen. 29 (1996) 3951-3962]. Finally I will describe the thermodynamics of the model as where calculated by Lou et al. [Phys. Rev. Lett. 85, 11 (2000)].
We study numerically the elementary excitation spectrum of the most general isotropic spin-1 chain with bilinear-biquadratic nearest-neighbor coupling. Using finite-size scaling, a massless phase with a period tripling in the ground state is found to exist in an extended region around the Lai-Sutherland point. The location of the transition from the known valence-bond-like phase to the trimerized phase cannot, however, be given precisely. We also present results on the ground-state two-point correlation function.
We investigate the ground-state phase diagram of the S=12\mathrm{S}=\frac{1}{2}S=21​ two-leg XXZXXZXXZ spin-ladder system with an isotropic interchain coupling. In this model, there is the Berezinskii-Kosterlitz-Thouless transition which occurs at the XYXYXY-Haldane and XYXYXY-rung singlet phase boundaries. It was difficult to determine the transition line using traditional methods. We overcome this difficulty using the level spectroscopy method combined with the twisted boundary condition method, and we check the consistency. We find out that the phase boundary between XYXYXY phase and Haldane phase lies on $\ensuremath{\Delta}=0$ line. And we show that there exist two different XYXYXY phases, which we can distinguish investigating a XXXXXX correlation function.
We investigate critical phenomena of a spin-1 chain in the vicinity of the SU(3) symmetric critical point, which we already specified in a previous study [Mashiko and Nomura, Phys. Rev. B 104, 155405 (2021)]. We numerically diagonalize a Hamiltonian combining the bilinear-biquadratic Hamiltonian with the trimer Hamiltonian. We then discuss the numerical results based on the conformal field theory and the renormalization group. As a result, we first verify that the critical point found in our previous study is the tricritical point among the Haldane phase, the trimer phase, and the the trimer-liquid (TL) phase. Second, with regard to the TL-trimer transition and the TL-Haldane transition, we find that the critical phenomena around this tricritical point belong to the Berezinskii-Kosterlitz-Thouless-like universality class. Third, we find the boundary between the Haldane phase and the trimer phase, which is illustrated by the massive self-dual sine-Gordon model.
We have studied the Spin 1 antiferromagnetic chain with both bilinear and biquadratic exchange interactions. The spin correlations have been calculated by means of the transfer matrix method based on the Trotter formula. The mass gap ( m (0)) at the ground state has been investigated as a function of the ratio λ between bilinear and biquadratic interactions. The result suggests that the gapless ground state persists in a finite range of λ, i.e., λ c < λ< λ 1 , with λ c = 0.5 and λ 1 > 1 which is consistent with the exact solution at the completely integrable point (λ= 1). The calculated mass gap ( m (0)) has been fitted to a Kosterlitz-Thouless (KT)-type for λ< λ c ( m (0)= A exp [- B /(λ c - λ) σ ]) producing the results σ= 0.49±0.04 and λ c =0.49±0.03.